A206435 -id:A206435 - cp's OEIS frontend

A066967 Total sum of odd parts in all partitions of n.

1, 2, 7, 10, 23, 36, 65, 94, 160, 230, 356, 502, 743, 1030, 1480, 2006, 2797, 3760, 5120, 6780, 9092, 11902, 15701, 20350, 26508, 34036, 43860, 55822, 71215, 89988, 113792, 142724, 179137, 223230, 278183, 344602, 426687, 525616, 647085, 792950

Offset: 1

Vladeta Jovovic, Jan 26 2002

nonn

Partial sums of A206435. - Omar E. Pol, Mar 17 2012
From Omar E. Pol, Apr 01 2023: (Start)
Convolution of A000041 and A000593.
Convolution of A002865 and A078471.
a(n) is also the sum of all odd divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned odd divisors are also all odd parts of all partitions of n. (End)

			a(4) = 10 because in the partitions of 4, namely [4],[3,1],[2,2],[2,1,1],[1,1,1,1], the total sum of the odd parts is (3+1)+(1+1)+(1+1+1+1) = 10.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
George E. Andrews and Mircea Merca, A further look at the sum of the parts with the same parity in the partitions of n, Journal of Combinatorial Theory, Series A, Volume 203, 105849 (2024).

Cf. A000041, A000593, A066897, A066898, A113685, A206435.

Cf. A002865, A078471.

g:=sum((2*i-1)*x^(2*i-1)/(1-x^(2*i-1)),i=1..50)/product(1-x^j,j=1..50): gser:=series(g,x=0,50): seq(coeff(gser,x^n),n=1..47);
# Emeric Deutsch, Feb 19 2006
b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+ (i mod 2)*g[1]*i]
      fi
    end:
a:= n-> b(n, n)[2]:
seq (a(n), n=1..50);
# Alois P. Heinz, Mar 22 2012

max = 50; g = Sum[(2*i-1)*x^(2*i-1)/(1-x^(2*i-1)), {i, 1, max}]/Product[1-x^j, {j, 1, max}]; gser = Series[g, {x, 0, max}]; a[n_] := SeriesCoefficient[gser, {x, 0, n}]; Table[a[n], {n, 1, max-1}] (* Jean-François Alcover, Jan 24 2014, after Emeric Deutsch *)
Map[Total[Select[Flatten[IntegerPartitions[#]], OddQ]] &, Range[30]] (* Peter J. C. Moses, Mar 14 2014 *)

a(n) = Sum_{k=1..n} b(k)*numbpart(n-k), where b(k)=A000593(k)=sum of odd divisors of k.
a(n) = sum(k*A113685(n,k), k=0..n). - Emeric Deutsch, Feb 19 2006
G.f.: sum((2i-1)x^(2i-1)/(1-x^(2i-1)), i=1..infinity)/product(1-x^j, j=1..infinity). - Emeric Deutsch, Feb 19 2006
a(n) = A066186(n) - A066966(n). - Omar E. Pol, Mar 10 2012
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)). - Vaclav Kotesovec, May 29 2018

More terms from Naohiro Nomoto and Sascha Kurz, Feb 07 2002

A206436 Total sum of even parts in the last section of the set of partitions of n.

Original entry on oeis.org

0, 2, 0, 8, 2, 18, 10, 42, 28, 80, 70, 162, 148, 290, 300, 530, 562, 918, 1020, 1570, 1780, 2602, 3022, 4286, 4992, 6858, 8110, 10872, 12888, 16962, 20178, 26134, 31138, 39728, 47412, 59848, 71312, 89072, 106176, 131440, 156400, 192164, 228330, 278616, 330502

Offset: 1

Omar E. Pol, Feb 12 2012

nonn

Also total sum of even parts in the partitions of n that do not contain 1 as a part.
From Omar E. Pol, Apr 09 2023: (Start)
Convolution of A002865 and A146076.
a(n) is also the total sum of even divisors of the terms in the n-th row of the triangle A336811.
a(n) is also the sum of even terms in the n-th row of the triangle A207378.
a(n) is also the sum of even terms in the n-th row of the triangle A336812. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Partial sums give A066966.

Cf. A002865, A135010, A138121, A138879, A146076, A206433, A206434, A206435, A207378, A336811, A336812.

b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    else g:= b(n, i-1); h:= `if`(i>n, [0, 0], b(n-i, i));
         [g[1]+h[1], g[2]+h[2] +((i+1) mod 2)*h[1]*i]
      fi
    end:
a:= n-> b(n, n)[2] -`if`(n=1, 0, b(n-1, n-1)[2]):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 16 2012

b[n_, i_] := b[n, i] = Module[{g, h}, Which[n == 0, {1, 0}, i < 1, {0, 0}, True, g = b[n, i-1]; h = If[i>n, {0, 0}, b[n-i, i]]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + Mod[i+1, 2]*h[[1]]*i}]]; a[n_] := b[n, n][[2]] - If[n == 1, 0, b[n-1, n-1][[2]]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Feb 16 2017, after Alois P. Heinz *)

G.f.: (Sum_{i>0} 2*i*x^(2*i)*(1-x)/(1-x^(2*i))) / Product_{i>0} (1-x^i). - Alois P. Heinz, Mar 16 2012

a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (24*sqrt(2*n)). - Vaclav Kotesovec, May 29 2018

More terms from Alois P. Heinz, Mar 16 2012

A206433 Total number of odd parts in the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 3, 3, 7, 9, 15, 19, 32, 40, 60, 78, 111, 143, 200, 252, 343, 437, 576, 728, 952, 1190, 1531, 1911, 2426, 3008, 3788, 4664, 5819, 7143, 8830, 10780, 13255, 16095, 19661, 23787, 28881, 34795, 42051, 50445, 60675, 72547, 86859, 103481, 123442, 146548

Offset: 1

Omar E. Pol, Feb 12 2012

nonn

From Omar E. Pol, Apr 07 2023: (Start)
Convolution of A002865 and A001227.
a(n) is also the total number of odd divisors of the terms in the n-th row of the triangle A336811.
a(n) is also the number of odd terms in the n-th row of the triangle A207378.
a(n) is also the number of odd terms in the n-th row of the triangle A336812. (End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Partial sums give A066897.

Cf. A001227, A002865, A006128, A135010, A138121, A138137, A182703, A206434, A206435, A206436, A207378, A336811, A336812.

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+ (i mod 2)*g[1]]
      fi
    end:
a:= n-> b(n, n)[2] -b(n-1, n-1)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Mar 22 2012

b[n_, i_] := b[n, i] = Module[{f, g}, If[n==0 || i==1, {1, n}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]]+g[[1]], f[[2]]+g[[2]] + Mod[i, 2]*g[[1]]}]]; a[n_] := b[n, n][[2]]-b[n-1, n-1][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 16 2017, after Alois P. Heinz *)

More terms from Alois P. Heinz, Mar 22 2012

A206434 Total number of even parts in the last section of the set of partitions of n.

Original entry on oeis.org

0, 1, 0, 3, 1, 6, 4, 13, 10, 24, 23, 46, 46, 81, 88, 143, 159, 242, 278, 404, 470, 657, 776, 1057, 1251, 1663, 1984, 2587, 3089, 3967, 4742, 6012, 7184, 9001, 10753, 13351, 15917, 19594, 23335, 28514, 33883, 41140, 48787, 58894, 69691, 83680, 98809, 118101

Offset: 1

Omar E. Pol, Feb 12 2012

nonn

From Omar E. Pol, Apr 07 2023: (Start)
Convolution of A002865 and A183063.
a(n) is also the total number of even divisors of the terms in the n-th row of the triangle A336811.
a(n) is also the number of even terms in the n-th row of the triangle A207378.
a(n) is also the number of even terms in the n-th row of the triangle A336812. (End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Partial sums give A066898.

Cf. A002865, A006128, A135010, A138121, A138137, A182703, A183063, A206433, A206435, A206436, A207378, A336811, A336812.

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+ ((i+1) mod 2)*g[1]]
      fi
    end:
a:= n-> b(n, n)[2] -b(n-1, n-1)[2]:
seq (a(n), n=1..50);  # Alois P. Heinz, Mar 22 2012

b[n_, i_] := b[n, i] = Module[{f, g}, If[n == 0 || i == 1, {1, 0}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + Mod[i+1, 2]*g[[1]]}]]; a[n_] := b[n, n][[2]]-b[n-1, n-1][[2]]; Table[ a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 16 2017, after Alois P. Heinz *)

G.f.: (Sum_{i>0} (x^(2*i)-x^(2*i+1))/(1-x^(2*i)))/Product_{i>0} (1-x^i). - Alois P. Heinz, Mar 23 2012

More terms from Alois P. Heinz, Mar 22 2012

cp's OEIS Frontend

A066967 Total sum of odd parts in all partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A206436 Total sum of even parts in the last section of the set of partitions of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A206433 Total number of odd parts in the last section of the set of partitions of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A206434 Total number of even parts in the last section of the set of partitions of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions